Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#21333 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C24 | C24×C7 |
| prime radical | C25×C7 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C7×C22⋊C4 ◁ G | |
1338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,14)(11,12)(13,15)(16,17)(18,21,25,28)(19,22)(20,23)(24,27,30,31)(26,29);(9,12)(13,14)(16,18)(17,21,23,28)(19,24)(20,25)(22,27,29,31)(26,30);(8,10,11,15)(9,13,12,14)(16,19)(17,22)(18,24)(20,26)(21,27)(23,29)(25,30)(28,31);(19,26)(22,29)(24,30)(27,31)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14321);